Incremental variance and/or standard deviation calculation for big data or streamed data using components

ABSTRACT

The present invention extends to methods, systems, and computing system program products for incrementally calculating variance and/or standard deviation for Big Data or streamed data. Embodiments of the invention include incrementally calculating one or more components of a variance and/or a standard deviation for a modified computation subset based on one or more components of the variance and/or the standard deviation calculated for a previous computation subset and then calculating variance and/or standard deviation based on the incrementally calculated components. Incrementally calculating the components of a variance and/or a standard deviation avoids visiting all data elements in the modified computation subset and performing redundant computations thereby increasing calculation efficiency, saving computing resources and reducing computing system&#39;s power consumption.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of provisional patent application Ser. No. 62/089,262, filed 2014 Dec. 9 by the present inventor.

BACKGROUND AND RELEVANT ART

Internet, mobile communications, navigation, online gaming, sensing technologies and large scale computing infrastructures are producing large amounts of data every day. Big Data is data that is beyond the processing capacity of conventional database systems and analyzing capacity of traditional analyzing methods due to its large volume and fast moving and growing speed. More companies now rely on Big Data to make real-time decisions to solve various problems. Current methods involve utilizing a lot of computational resources, which are very costly, yet still may not satisfy the needs of real-time decision making based on the newest information, especially in the financial industry. How to efficiently, promptly and cost-effectively process and analyze Big Data presents a difficult challenge to data analysts and computer scientists.

Streamed data is data that is constantly being received by a receiver while being delivered by a provider. Streamed data may be real-time data gathered from sensors and continuously transferred to computing devices or electronic devices. Often this includes receiving similarly formatted data elements in succession separated by some time interval. Streamed data may also be data continuously read from storage devices, e.g., storage devices on multi-computing devices which store a Big Data set. Stream processing has become a focused research area recently due to the following reasons. One reason is that the input data are coming too fast to store entirely for batch processing, so some analysis have to be performed when the data streams in. The second reason is that immediate responses to any changes of the data are required in some application domains, e.g., mobile related applications, online gaming, navigation, real-time stock analysis and automated trading, etc. The third reason is that some applications or electronic devices require streaming processing due to their nature, e.g., audio, video and digital TV, etc.

In addition, methods on streamed data processing may be extended to Big Data processing, because Big Data sets are accumulated over time and they may be considered as a data stream with irregular time intervals.

Processing Big Data or streamed data may include performing calculations on multiple data elements. When performing statistical calculations on streamed data elements, the number of data elements to be accessed may be quite large. For example, when calculating a variance and/or a standard deviation a (potentially large) number of data elements may need to be accessed.

Further, some statistical calculations are recalculated as new data elements are added to a Big Data set or new streamed data elements are received. Thus, the (potentially large) number of data elements may be repeatedly accessed. For example, it may be that a variance and/or a standard deviation is calculated for a computation subset whose size n keeps increasing to include the newly accessed or received data elements. As such, every time a new data element is accessed or received, the new element is added to the computation subset. The n+1, n+2, . . . data elements in the computation subset are then accessed to recalculate the variance and/or the standard deviation.

When performing a variance and/or a standard deviation calculation on n+1 data elements using traditional algorithms, all the n+1 data elements in the computation subset will be visited and used. Depending on necessity, the computation subset length n may be extremely large, so the data elements in a computation subset may be distributed over a cloud comprising hundreds of thousands of computing devices. Re-performing a variance and/or a standard deviation calculation on Big Data or streamed data elements in traditional way is time consuming and is an inefficient use of resources.

BRIEF SUMMARY

The present disclosure describes methods, systems, and computing system program products for incrementally calculating a variance and/or a standard deviation for Big Data or streamed data. A computing system comprises one or more computing devices. Each of the computing devices comprises one or more processors. The computing system comprises one or more storage media. The computing system has stored a data set on the one or more storage media or has access to a data stream. The computing system maintains a computation subset size counter. The computation subset size counter indicates the number of data elements in a computation subset of the data set or the data stream. Embodiments of the invention include incrementally calculating one or more components of a variance and/or a standard deviation for a modified computation subset based on one or more components calculated for the previous computation subset and then calculating the variance and/or the standard deviation for the modified computation subset based on one or more of the incrementally calculated components. Incrementally calculating a variance and/or a standard deviation not only avoids visiting all the data element in the computation subset but also avoids storing the whole computation subset as traditional algorithms would do, and thus improves the calculation efficiency, saves computing resources and reduces computing system's power consumption. The computing system may also optionally include a data buffer for storing accessed or received data elements.

The computing system incrementally calculates one or more components of a variance and/or a standard deviation starting from either an empty computation subset or a non-empty computation subset if one or more components have already been calculated.

When incremental variance and standard deviation calculation starts from an empty computation subset, the computation subset size is initialized with a zero and one or more components are initialized with zero values.

The computing system initializes a computation subset size counter and one or more (v(1≤v≤p)) components of a variance and/or a standard deviation.

The computing system accesses or receives a Big Data or streamed data element to be added to a computation subset.

The computing system stores the accessed or received data element into an optional data buffer as needed.

The computing system modifies the computation subset by adding the accessed or received data element to the computation subset and modifies the computation subset size by increasing its value by 1.

The computing system directly incrementally calculates one or more (v(1≤v≤p)) components of a variance and/or a standard deviation for the modified computation subset. Directly incrementally calculating the v components of a variance and/or a standard deviation includes directly incrementally calculating each of the v components one by one. Directly incrementally calculating a component includes accessing the v component calculated for the prior computation subset and adding a contribution of the added data element to the component mathematically.

The computing system indirectly incrementally calculating w=p−v components as needed: indirectly incrementally calculating the w components includes indirectly incrementally calculating each of the w components one by one. Indirectly incrementally calculating a component includes calculating the component based on one or more components other than the component itself. The one or more components may have been initialized, directly incrementally calculated or indirectly incrementally calculated.

The computing system generates a variance and/or a standard deviation as needed based on one or more incrementally calculated components.

The computing system may keep accessing or receiving a data element to be added to the computation subset, storing the accessed or received data element into an optional data buffer as needed, modifying the computation subset and the computation subset size, directly incrementally calculating v(1≤v≤p) components, indirectly incrementally calculating w=p−v components as needed and generating a variance and/or a standard deviation as needed, and the computing system may repeat this process for as many times as needed.

This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter.

Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by the practice of the invention. The features and advantages of the invention may be realized and obtained by means of the instruments and combinations particularly pointed out in the appended claims. These and other features of the present invention will become more fully apparent from the following description and appended claims, or may be learned by the practice of the invention as set forth hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to describe the manner in which the above-recited and other advantages and features of the invention may be obtained, a more particular description of the invention briefly described above will be rendered by reference to specific embodiments thereof which are illustrated in the appended drawings. Understanding that these drawings depict only typical embodiments of the invention and are not therefore to be considered to be limiting of its scope, the invention will be described and explained with additional specificity and detail through the use of the accompanying drawings in which:

FIG. 1 illustrates a high-level overview of an example computing system that facilitates incrementally calculating variance and/or standard deviation for Big Data or streamed data.

FIG. 1A illustrates an example computing system architecture that facilitates incrementally calculating a variance and/or a standard deviation for streamed data with all components being directly incrementally calculated.

FIG. 1B illustrates an example computing system architecture that facilitates incrementally calculating a variance and/or a standard deviation for streamed data with some components being directly incrementally calculated and some components being indirectly incrementally calculated.

FIG. 1C illustrates an example computing system architecture that facilitates incrementally calculating a variance and/or a standard deviation for Big Data with all components being directly incrementally calculated.

FIG. 1D illustrates anb example computing system architecture that facilitates incrementally calculating a variance and/or a standard deviation for Big Data with some components being directly incrementally calculated and some components being indirectly incrementally calculated.

FIG. 2 illustrates a flow chart of an example method for incrementally calculating a variance and/or a standard deviation for Big Data or streamed data.

FIG. 3A illustrates data that is accessed from a computation window for incrementally calculating a variance and/or a standard deviation on streamed data.

FIG. 3B illustrates data that is accessed from a computation subset for incrementally calculating a variance and/or a standard deviation on Big Data.

FIG. 4A illustrates the definitions of a sample variance, a population variance, a sample standard deviation and a population standard deviation and traditional equations for calculating them.

FIG. 4B illustrates some example components of a variance and a standard deviation and a few basic incremental component calculation equations commonly used by different incremental methods.

FIG. 4C illustrates the first example incremental variance and/or standard deviation calculation algorithm (incremental algorithm 1) and its equations for incrementally calculating a sample variance and a sample standard deviation based on incrementally calculated sample variance, and a population variance and a population standard deviation based on incrementally calculated population variance respectively based on the incrementally calculated mean x _(k+1).

FIG. 4D illustrates the second example variance and/or standard deviation calculation algorithm (incremental algorithm 2) and its equations for incrementally calculating a sample variance, a population variance, a sample standard deviation and a population standard deviation respectively in a computation subset based on incrementally calculated

${SSD}_{k + 1} = {{\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k + 1}} \right)^{2}} + \left( {x_{a} - {\overset{\_}{x}}_{k + 1}} \right)^{2}}$ and mean x _(k+1).

FIG. 4E illustrates the third example variance and/or standard deviation calculation algorithm (incremental algorithm 3) and its equations for incrementally calculating a sample variance, a population variance, a sample standard deviation and a population standard deviation respectively in a computation subset based on incrementally calculated

${SS}_{k + 1} = {{\sum\limits_{1}^{n}x_{i}^{2}} + x_{a}^{2}}$ and mean x _(k+1).

FIG. 5A illustrates an example of calculating sample variance and sample standard deviation using traditional algorithms as shown in FIG. 4A.

FIG. 5B illustrates an example of calculating sample variance and sample standard deviation using incremental algorithm 1 as shown in FIG. 4C.

FIG. 5C illustrates an example of calculating sample variance and sample standard deviation using incremental algorithm 2 as shown in FIG. 4D.

FIG. 5D illustrates an example of calculating sample variance and sample standard deviation using incremental algorithm 3 as shown in FIG. 4E.

FIG. 6 illustrates computational loads for traditional sample variance algorithms and incremental sample variance algorithms with a computation subset of size 10.

FIG. 7 illustrates computational loads for traditional sample variance algorithms and incremental sample variance algorithms with a computation subset of size 1,000,000.

FIG. 8 illustrates computational loads for traditional sample standard deviation algorithms and incremental sample standard deviation algorithms with a computation subset of size 10.

FIG. 9 illustrates computational loads for traditional sample standard deviation algorithms and incremental sample standard deviation algorithms with a computation subset of size 1,000,000.

DETAILED DESCRIPTION

The present disclosure describes methods, systems, and computing system program products for incrementally calculating a variance and/or a standard deviation for Big Data or streamed data. A computing system comprises one or more computing devices. Each of the computing devices comprises one or more processors. The computing system comprises one or more storage media. The computing system has stored a data set on the one or more storage media or has access to a data stream. The computing system maintains a computation subset size counter. The computation subset size counter indicates the number of data elements in a computation subset of the data set or the data stream. Embodiments of the invention include incrementally calculating one or more components of a variance and/or a standard deviation for a modified computation subset based on one or more components calculated for the previous computation subset and then calculating the variance and/or the standard deviation for the modified computation subset based on one or more of the incrementally calculated components. Incrementally calculating a variance and/or a standard deviation not only avoids visiting all the data element in the computation subset but also avoids storing the whole computation subset as traditional algorithms would do, and thus improves the calculation efficiency, saves computing resources and reduces computing system's power consumption. The computing system may also optionally include a data buffer for storing accessed or received data elements.

A variance of a random variable is a measure for the width of its distribution which describes how much its values disperse around the expected value. A standard deviation of a random variable is the square-root of the variance of the random variable and it is a measure for the scale of a distribution.

A computation subset is a subset of a Big Data set which contains the data elements involved in a variance and/or a standard deviation calculation. A computation subset is equivalent to a moving computation window when performing a variance and/or a standard deviation calculation on streamed data or time series data. Within the description of embodiments of the present invention, the difference between a computation window and a computation subset is that data elements in a computation window are ordered but that in a computation subset are not.

As used herein, a component of a variance is a quantity or expression appearing in the variance's definition equation or any transforms of the equation, and a component of a standard deviation is a quantity or expression appearing in the standard deviation's definition equation or any transforms of the equation. A variance comprises one or more components. A variance itself may be considered as the largest component of a variance. A standard deviation also comprises one or more components including the standard deviation itself. A variance may be calculated using its components. Similarly, a standard deviation may be calculated using its components. A standard deviation for a data set is the square root of a variance for the same data set, so they share some common components. Some example components of a variance and/or a standard deviation may be found in FIG. 4B.

A component may be either directly incrementally calculated or indirectly incrementally calculated. The difference between them is that when directly incrementally calculating a component, the component's value from previous iteration is used as a part of the calculation but when indirectly incrementally calculating a component, one or more components other than the component itself are used.

For a given component, it might be directly incrementally calculated in one algorithm but indirectly incrementally calculated in another algorithm.

For a given algorithm, assume the total number of different components is p(p≥1), the number of directly incrementally calculated components is v(1≤v≤p), then the number of indirectly incrementally calculated components is w=p−v(0≤w<p). For any algorithm, there will be at least one component being directly incrementally calculated. It is possible that all components are directly incrementally calculated (in this case v=p and w=0). However, directly incrementally calculated components must be calculated in every iteration no matter if a variance and/or a standard deviation is accessed or not in a specific iteration.

For a given algorithm, if a component is directly incrementally calculated, then the component must be calculated in every iteration (i.e., whenever a data element is added to the computation subset). However, if a component is indirectly incrementally calculated, then the component only needs to be calculated as needed (i.e., when a variance and/or a standard deviation need(s) to be calculated and accessed). Thus, when a variance and/or a standard deviation are not accessed in a specific iteration, only a small number of components are incrementally calculated to save computation time. It should be understood that an indirectly incrementally calculated component may also be used in the calculation of a directly incrementally calculated component. In that case, the indirectly incrementally calculated component should also be calculated in every iteration.

Embodiments of the invention include incrementally calculating each of the one or more (p(p≥1)) components of a variance and/or a standard deviation in a modified computation subset based on one or more (v(1≤v≤p)) components calculated for a previous computation subset.

The computing system incrementally calculates one or more components of a variance and/or a standard deviation starting from either an empty computation subset or a non-empty computation subset if one or more components have already been calculated.

The computing system initializes a computation subset size counter and one or more (v(1≤v≤p)) components of a variance and/or a standard deviation.

When the computing system incrementally calculates the v components of a variance and/or a standard deviation starting from an empty computation subset, the computation subset size is initialized with a zero and the v components are initialized with zero values.

When incremental variance and standard deviation calculation starts from a non-empty computation subset, the computation subset size and one or more components are initialized. The initialization of the computation subset size comprises counting the number of data elements contained in the computation subset or accessing or receiving a predefined computation subset size. The initialization of the one or more components comprises calculating the one or more components through their definitions based on the data elements in the computation subset or accessing or receiving pre-calculated one or more components from one or more computing-device-readable media.

The computing system accesses or receives a data element to be added to the computation subset.

The computing system stores the accessed or received data element into an optional data buffer as needed. Pure incremental variance and standard deviation calculation only need access to the data element just accessed or received but don't need access to data elements accessed or received earlier than the just accessed or received data element, so the computing system doesn't necessarily allocate a storage space to keep all the data elements of a computation subset but at least always keeps the computation subset size and one or more components of a variance and/or a standard deviation for the computation subset, and modifying a computation subset may be reflected by modifying the computation subset size and one or more components of a variance and/or a standard deviation for the computation subset. However, since iterative variance and/or standard deviation calculation requires access to earlier accessed or received data elements, when combining incremental variance and/or standard deviation calculation with iterative variance and/or standard deviation calculation, the accessed or received data element may need to be stored into a data buffer for future usage. Thus, the computing system may optionally have a data buffer for storing the accessed or received data elements for the computation subset and may need to store the accessed or received data element into the optional data buffer.

The computing system modifies the computation subset by adding the accessed or received data element to the computation subset and modifies the computation subset size by increasing its value by 1.

The computing system directly incrementally calculates one or more v(1≤v≤p) components of a variance and/or a standard deviation for the modified computation subset. Directly incrementally calculating the v components of a variance and/or a standard deviation includes directly incrementally calculating each of the v components one by one. Directly incrementally calculating a component includes accessing the component calculated for the prior computation subset and adding a contribution of the added data element to the component mathematically.

The computing system indirectly incrementally calculating w=p−v components as needed: indirectly incrementally calculating the w components includes indirectly incrementally calculating each of the w components one by one. Indirectly incrementally calculating a component includes calculating the component using one or more components other than the component itself (Depending on a specific algorithm used, calculating each of the w components may also need to access and use of the data element added to the computation subset). The one or more components may have been initialized, directly incrementally calculated or indirectly incrementally calculated.

The computing system generates a variance and/or a standard deviation as needed based on one or more incrementally calculated components.

The computing system may keep accessing or receiving a data element to be added to the computation subset, storing the data element into an optional data buffer as needed, modifying the computation subset and the computation subset size, directly incrementally calculating v(1≤v≤p) components, indirectly incrementally calculating w=p−v components as needed and generating a variance and/or a standard deviation as needed, and the computing system may repeat this process for as many times as needed.

Embodiments of the present invention may comprise or utilize a special purpose or general-purpose computing device including computing device hardware, such as, for example, one or more processors and system memory, as discussed in greater detail below. Embodiments within the scope of the present invention also include physical and other computing-device-readable media for carrying or storing computing-device-executable instructions and/or data structures. Such computing-device-readable media may be any available media that may be accessed by a general purpose or special purpose computing device. Computing-device-readable media that store computing-device-executable instructions are computing device storage media (devices). Computing-device-readable media that carry computing-device-executable instructions are transmission media. Thus, by way of example, and not limitation, embodiments of the invention may comprise at least two distinctly different kinds of computing-device-readable media: computing device storage media (devices) and transmission media.

Computing device storage media (devices) includes RAM, ROM, EEPROM, CD-ROM, solid state drives (“SSDs”) (e.g., based on RAM), Flash memory, phase-change memory (“PCM”), other types of memory, other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which may be used to store desired program code means in the form of computing-device-executable instructions or data structures and which may be accessed by a general purpose or special purpose computing device.

A “network” is defined as one or more data links that enable the transport of electronic data between computing devices and/or modules and/or other electronic devices. When information is transferred or provided over a network or another communications connection (either hardwired, wireless, or a combination of hardwired or wireless) to a computing device, the computing device properly views the connection as a transmission medium. Transmissions media may include a network and/or data links which may be used to carry desired program code means in the form of computing-device-executable instructions or data structures and which may be accessed by a general purpose or special purpose computing device. Combinations of the above should also be included within the scope of computing-device-readable media.

Further, upon reaching various computing device components, program code means in the form of computing-device-executable instructions or data structures may be transferred automatically from transmission media to computing device storage media (devices) (or vice versa). For example, computing-device-executable instructions or data structures received over a network or data link may be buffered in RAM within a network interface module (e.g., a “NIC”), and then eventually transferred to computing device RAM and/or to less volatile computing device storage media (devices) at a computing device. Thus, it should be understood that computing device storage media (devices) may be included in computing device components that also (or even primarily) utilize transmission media.

Computing-device-executable instructions comprise, for example, instructions and data which, when executed at a processor, cause a general purpose computing device or special purpose computing device to perform a certain function or group of functions. The computing device executable instructions may be, for example, binaries, intermediate format instructions such as assembly language, or even source code. Although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the described features or acts described above. Rather, the described features and acts are disclosed as example forms of implementing the claims.

Those skilled in the art will appreciate that embodiments of the present invention may be practiced in network computing environments with many types of computing device configurations, including, personal computers, desktop computers, laptop computers, message processors, hand-held devices, multi-processor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, supercomputers, mobile telephones, PDAs, tablets, pagers, routers, switches, and the like. Embodiments of the present invention may also be practiced in distributed system environments where local and remote computing devices, which are linked (either by hardwired data links, wireless data links, or by a combination of hardwired and wireless data links) through a network, both perform tasks. In a distributed system environment, program modules may be located in both local and remote memory storage devices.

Embodiments of the invention may also be implemented in cloud computing environments. In this description and the following claims, “cloud computing” is defined as a model for enabling on-demand network access to a shared pool of configurable computing resources. For example, cloud computing may be employed in the marketplace to offer ubiquitous and convenient on-demand access to the shared pool of configurable computing resources. The shared pool of configurable computing resources may be rapidly provisioned via virtualization and released with low management effort or service provider interaction, and then scaled accordingly.

A cloud computing model may be composed of various characteristics such as, for example, on-demand self-service, broad network access, resource pooling, rapid elasticity, measured service, and so forth. A cloud computing model may also expose various service models, such as, for example, Software as a Service (“SaaS”), Platform as a Service (“PaaS”), and Infrastructure as a Service (“IaaS”). A cloud computing model may also be deployed using different deployment models such as private cloud, community cloud, public cloud, hybrid cloud, and so forth. In this description and in the claims, a “cloud computing environment” is an environment in which cloud computing is employed.

FIG. 1 illustrates a high-level overview of an example computing system 100 that facilitates incrementally calculating variance and/or standard deviation for Big Data or streamed data. Referring to FIG. 1, computing system 100 comprises multiple devices connected by different networks, such as local network, internet and wireless network, etc. The multiple devices include, for example, a data analysis engine 1007, a storage system 1011, live data stream 1006, and multiple distributed computing devices that may schedule data analysis tasks and/or query data analysis results, such as personal computer 1016, hand-held devices 1017 and desktop computer 1018, etc. Data analysis engine 1007 may comprise one or more processors, e.g., CPU 1009 and CPU 1010, one or more system memory, e.g., system memory 1008, variance calculation module 191, standard deviation calculation 193 and component calculation modules and 131. Variance calculation module 191, standard deviation calculation 193 and component calculation modules 131 will be illustrated in more details in other figures. Storage system 1011 may comprise one or more storage media, e.g., storage medium 1012 and storage medium 1014, which may be used for hosting Big Data sets. Data sets on storage system 1011 may be accessed by data analysis engine 1007. In general, data stream 1006 may comprise streamed data from different data sources, for example, stock quotes, audio data, video data, geospatial data, web data, mobile communication data, online gaming data, banking transaction data, sensor data, closed-captioning data, etc. To depict a few, real-time data 1000 may comprise data collected from sensor 1001, stock 1002, web 1003 and bank 1004, etc. in real-time. Data analysis engine 1007 may receive data elements from data stream 1006. Data from different data sources may also be stored in storage system 1011 which may be accessed for Big Data analysis. It should be understood that FIG. 100 is provided to introduce a selection of concepts in a much simplified form, for example, distributed devices 1016 and 1017 may need to go through a firewall to connect data analysis engine 1007, and data accessed or received from data stream 1006 and/or storage system 1011 by data analysis engine 1007 may be filtered by data filters, etc.

FIG. 1A illustrates an example computing system architecture 100A that facilitates incrementally calculating a variance and/or a standard deviation for streamed data with all components being directly incrementally calculated, in other words p=v≥1 and w=0. FIG. 1A illustrates 1007 and 1006 shown in FIG. 1. Referring to FIG. 1A, computing system architecture 100A includes component calculation module 131, variance calculation module 191 and standard deviation calculation module 193. Component calculation module 131 may be connected to (or is part of) a network, such as, for example, a Local Area Network (“LAN”), a Wide Area Network (“WAN”), and even the Internet. Accordingly, component calculation module 131 as well as any other connected computing devices and their components, may send and receive message related data (e.g., Internet Protocol (“IP”) datagrams and other higher layer protocols that utilize IP datagrams, such as, User Datagram Protocol (“UDP”), Real-time Streaming Protocol (“RTSP”), Real-time Transport Protocol (“RTP”), Microsoft® Media Server (“MMS”), Transmission Control Protocol (“TCP”), Hypertext Transfer Protocol (“HTTP”), Simple Mail Transfer Protocol (“SMTP”), etc.) over the network. The output of component calculation module 131 may be used as the input of variance calculation module 191, and variance calculation module 191 may generate variance 192. The output of component calculation module 131 may be used as the input of standard deviation calculation module 193, and standard deviation calculation module 193 may generate standard deviation 194.

For example, as depicted in FIG. 1A, when data element 101 is received, the counter 118 will be increased by 1. Both the computation subset size counter 118 and data element 101 may be accessed or received by component calculation module 131.

Subsequently, data element 102 may be received. When this happens, the computation subset counter 118 will increase its value by 1. Both the modified computation subset size counter 118 and data element 102 may be accessed or received by component calculation module 131.

As streamed data elements are accessed or received, they may be stored into an optional data buffer 125 as needed (i.e., when combining incremental variance and/or standard deviation calculation with iterative variance and/or standard deviation calculation).

In general, data stream 190 may be a sequence of digitally encoded signals (e.g., packets of data or data packets) used to transmit or receive information that is in the process of being transmitted. Data stream 190 may stream data elements to computing system architecture 100A. Data stream 190 may stream stored data or be a live stream.

Computation subset size counter 118 may be implemented in hardware or software. When it is implemented in hardware, counter 118 may be a small device that may update a computation subset size stored in a location in counter 118. When it is implemented in software, counter 118 may be just a variable stored in RAM, hard drive or any other computing-device-readable storage media, and the content of counter 118 is a computation subset size. Computation subset size counter 118 may be used to keep track the size of a computation subset. Whenever receiving a data element, the computing system adjusts the computation subset by adding the data element to the computation subset and adjusts the computation subset size counter 118 by increasing its content or value by 1. Computation subset size counter 118 may be accessed or received by component calculation module 131, and it may be reset to 0 when incremental variance and/or standard deviation calculation is reset or set to a specific value by reset module 129 when incremental variance and/or standard deviation calculation starts working on a non-empty computation subset. Within the description of this disclosure, a computation subset size counter is equivalent to a computation subset size and may be used interchangeably.

Incremental algorithms are also configured to directly incrementally calculate v components for data elements in a computation subset. Incremental algorithm 133 accesses or receives a prior component Cd₁ value and an added data element as input. Incremental algorithm 133 directly incrementally calculates a component Cd₁ for the modified computation subset based on the prior component Cd₁ value and the added data element. Contribution addition module 133A may add a contribution for the added data element to the prior component Cd₁. Adding a contribution for the added data element may be used to directly incrementally calculate component Cd₁ for the modified computation subset. Incremental algorithm 139 works in a similar way as incremental algorithm 133. Incremental algorithm 139 accesses or receives a prior component Cd_(v) value and an added data element as input. Incremental algorithm 139 calculates a component Cd_(v) for the modified computation subset based on the prior component Cd_(v) value and the added data element. Contribution addition module 139A may add a contribution for the added data element to the prior component Cd_(v). Adding a contribution for the added data element may be used to directly incrementally calculate component Cd_(v) for the modified computation subset.

Referring to computing system architecture 100A, component calculation module 131 comprises v(v=p≥1) component calculation modules for directly incrementally calculating v components for data elements in a computation subset. The number of components v varies depending on which incremental algorithm is used. As depicted in FIG. 1A, component calculation module 131 comprises component Cd₁ calculation module 161 and component Cd_(v) calculation module 162, and there are v−2 other component calculation modules between them. Component Cd₁ calculation module 161 comprises initialization module 132 for initializing component Cd₁ and incremental algorithm 133 for incrementally calculating component Cd₁. Component Cd_(v) calculation module 162 comprises initialization module 138 for initializing component Cd_(v) and incremental algorithm 139 for incrementally calculating component Cd_(v). Initialization module 132 is configured to initialize component Cd₁ for data elements in a computation subset and Component Cd_(v) initialization module 138 is configured to initialize component Cd_(v) for data elements in a computation subset. Component Cd₁ 141 is the initial value of component Cd₁. Initialization module 132 may be used for an initialization of component Cd₁ or when variance and/or standard deviation calculations are reset. Initialization module 132 either initialize component Cd₁ 141 to be zero if the computation subset is empty (the computation subset size counter is zero) or a specific value contribution 151 passed in by reset module 129 or calculated from the data elements in the computation subset if the computation subset is non-empty. Similarly, initialization module 138 may be used for an initialization of component Cd_(v) or when variance and/or standard deviation calculations are reset. Component Cd_(v) 145 is the initial value of component Cd_(v). Initialization module 138 either initialize component Cd_(v) 145 to be zero if the computation subset is empty (the computation subset size counter is zero) or a specific value contribution 181 passed in by reset module 129 or calculated from the data elements in the computation subset if the computation subset is non-empty.

Referring to FIG. 1A, computing system architecture 100A also includes variance calculation module 191. Once p=v≥1 components of a variance and/or a standard deviation are calculated by component calculation module 131, variance calculation module 191 may calculate variance 192 as needed based on one or more incrementally calculated components. Variance 192 may be either a sample variance or a population variance.

Referring to FIG. 1A, computing system architecture 100A includes standard deviation calculation module 193. Once p=v≥1 components of a variance and/or a standard deviation are calculated by component calculation module 131, standard deviation calculation module 193 may calculate standard deviation 194 as needed using one or more incrementally calculated components. Standard deviation 194 may be either a sample standard deviation or a population standard deviation.

FIG. 1B illustrates an example computing system architecture 100B that facilitates incrementally calculating variance and/or standard deviation for Big Data or streamed data with some (v(1≤v<p)) components being directly incrementally calculated and some (w(w=p−v)) components being indirectly incrementally calculated. The number v as well as the number w is algorithm dependent. Many parts included in computing system architectures 100B and 100A have same reference numbers. Those parts have similar structures and work in similar ways. In certain implementations, the difference between the computing system architectures 100B and 100A may be that architecture 100B includes a component calculation module 135. All parts except component calculation module 135 in 100B work in a similar way as those parts with same reference numbers in 100A. Instead of repeating what have already been explained in the description about 100A, only the different part is discussed here. The number v in 100B may not be the same number v as in 100A, because some directly incrementally calculated components in 100A are indirectly incrementally calculated in 100B. In 100A, v=p≥1, but in 100B, 1≤v<p. Referring to FIG. 1B, computing system architecture 100B includes component calculation module 135. The output of components calculation module 131 may be used as the input of component calculation module 135, and the output of calculation modules 131 and 135 may be used as the input of variance calculation module 191, and variance calculation module 191 may generate variance 192. The output of calculation modules 131 and 135 may be used as the input of standard deviation calculation module 193, and standard deviation calculation module 193 may generate standard deviation 194. Component calculation module 135 generally includes w=p−v component calculation modules for indirectly incrementally calculating w components. For example, component calculation module 135 includes calculation module 163 for indirectly incrementally calculating component Ci₁ and calculation module 164 for indirectly incrementally calculating component Ci_(w), and there are w−2 component calculation modules between them. Indirectly incrementally calculating w components includes indirectly incrementally calculating each of the w components one by one. Indirectly incrementally calculating a component includes accessing and based on one or more components other than the component itself. The one or more components may have been initialized, directly incrementally calculated or indirectly incrementally calculated.

Referring to computing system architecture 100B, once all p(p=v+w) components have been calculated, variance calculation module 191 may be used for calculating a variance 192 as needed and standard deviation calculation module 193 may be used for calculating a standard deviation 194 as needed based on one or more incrementally calculated components. Variance 192 may be either a sample variance or a population variance. Standard deviation 194 may be either a sample standard deviation or a population standard deviation.

FIG. 1C illustrates an example computing system architecture 100C that facilitates incrementally calculating variance and/or standard deviation for streamed data with all components (p(p=v≥1)) being directly incrementally calculated, in other words p=v≥1 and w=0. FIG. 1C illustrates 1007 and 1006 shown in FIG. 1. Referring to FIG. 1C, computing system architecture 100C includes component calculation module 131, variance calculation module 191 and standard deviation calculation module 193. Component calculation module 131 may be tightly coupled with one or more storage media by a high-speed data bus or loosely coupled with one or more storage media managed by a storage system via (or is part of) a network, such as, for example, a Local Area Network (“LAN”), a Wide Area Network (“WAN”), and even the Internet.

In general, storage media 121 may be a single local storage medium and may also be a complex storage system that comprises multiple physically distributed storage devices managed by a storage management system.

Storage media 121 comprises a data set 123. As depicted, data set 123 comprises multiple data elements stored in multiple locations of storage media 121. For example, data elements 106, 105, 103, 104, 101, and 102 are stored in locations 121A, 121B, 121C, 121D, 121E and 121F, etc., . . . . There are multiple data elements stored in other locations of storage media 121.

Assume that a variance and/or standard deviation calculation is started from computation subset 122. Computation subset 122 with a specified size of 4 (i.e., n=4) contains a number of data elements of data set 123: data elements 106, 105, 103 and 104. All the data elements ranging from 103 to 106 in computation subset 122 may be accessed for initializing one or more components of the variance and/or standard deviation.

Next, data elements 101 may be accessed from locations 121E. Data element 101 will be added to computation subset 122, then computation subset 122 becomes modified computation subset 122A. Modified computation subset 122A contains 5 data elements: 106, 105, 103, 104 and 101. The computing system may incrementally calculate a variance and/or standard deviation for modified computation subset 122A.

Subsequently, data element 102 may be accessed from locations 121F. Data element 102 will be added to modified computation subset 122A, then modified computation subset 122A becomes modified computation subset 122B. Modified computation subset 122B contains 6 data elements: 106, 105, 103, 104, 101 and 102. The computing system may incrementally calculate a variance and/or standard deviation for modified computation subset 122B.

Referring to computing system architecture 100C, component calculation module 131 comprises v(v=p≥1) component calculation modules for directly incrementally calculating v components for data elements in a computation subset. The number of components, v, varies depending on which incremental algorithm is used. As depicted in FIG. 1C, component calculation module 131 comprises component Cd₁ calculation module 161 and component Cd_(v) calculation module 162, and there are v−2 other component calculation modules between them. Component Cd₁ calculation module 161 comprises initialization module 132 for initializing component Cd₁ and incremental algorithm 133 for incrementally calculating component Cd₁. Component Cd_(v) calculation module 162 comprises initialization module 138 for initializing component Cd_(v) and incremental algorithm 139 for incrementally calculating component Cd_(v). Initialization module 132 is configured to initialize component Cd₁ for data elements in a computation subset and initialization module 138 is configured to initialize component Cd_(v) for data elements in a computation subset. Component Cd₁ 141 is the initial value of component Cd₁. Initialization module 132 may be used for an initialization of component Cd₁ or when variance and/or standard deviation calculations are reset. Initialization module 132 initializes component Cd₁ 141 to be zero if the computation subset is empty (the computation subset size counter is zero) or a specific value contribution 151 passed in by reset module 129 or a value calculated from the data elements in the computation subset if the computation subset is non-empty. Similarly, initialization module 138 may be used for an initialization of component Cd_(v) or when variance and/or standard deviation calculations are reset. Component Cd_(v) 145 is the initial value of component Cd_(v). Initialization module 138 initializes component Cd_(v) 145 to be zero if the computation subset is empty (the computation subset size counter is zero) or a specific value contribution 181 passed in by reset module 129 or a value calculated from the data elements in the computation subset if the computation subset is non-empty.

Incremental algorithms are also configured to directly incrementally calculate v components for data elements in a computation subset. Incremental algorithm 133 receives a prior component Cd₁ value and an added data element as input. Incremental algorithm 133 directly incrementally calculates a component Cd₁ for the modified computation subset based on the prior component Cd₁ value and the added data element. Contribution addition module 133A may add a contribution for the added data element to the prior component Cd₁. Adding a contribution for the added data element may be used to calculate component Cd₁ for the modified computation subset. Incremental algorithm 139 works in a similar way as incremental algorithm 133. Incremental algorithm 139 receives a prior component Cd_(v) value and an added data element as input. Incremental algorithm 139 directly incrementally calculates a component Cd_(v) for the modified computation subset based on the prior component Cd_(v) value and the added data element. Contribution addition module 139A may add a contribution for the added data element to the prior component Cd_(v). Adding a contribution for the added data element may be used to calculate component Cd_(v) for the modified computation subset.

Referring to FIG. 1C, computing system architecture 100C also includes variance calculation module 191. Once p=v≥1 components of a variance and/or a standard deviation are calculated by component calculation module 131, variance calculation module 191 may calculate variance 192 as needed based on one or more incrementally calculated components. Variance 192 may be either a sample variance or a population variance.

Referring to FIG. 1C, computing system architecture 100C includes standard deviation calculation module 193. Once p=v≥1 components of a variance and/or a standard deviation are calculated by component calculation module 131, standard deviation calculation module 193 may calculate standard deviation 194 as needed using one or more incrementally calculated components. Standard deviation 194 may be either a sample standard deviation or a population standard deviation.

FIG. 1D illustrates an example computing system architecture 100D that facilitates incrementally calculating variance and/or a standard deviation for Big Data with some (v(1≤v<p)) components being directly incrementally calculated and some (w(w=p−v)) components being indirectly incrementally calculated. Number v and number w are algorithm dependent. Many parts included in computing system architectures 100D and 100C have same reference numbers. Those parts have similar structures and work in similar ways. In certain implementations, the difference between computing system architectures 100D and 100C may be that architecture 100D includes a component calculation module 135. All parts except component calculation module 135 in 100D work in a similar way as those parts with the same reference numbers in 100C. Instead of repeating what have already been explained in the description about 100C, only the different part is discussed here. The number v in 100D may not be the same number v as in 100C, because some directly incrementally calculated components in 100C are indirectly incrementally calculated in 100D. In 100C, v=p≥1, but in 100D, 1≤v<p. Referring to FIG. 1D, computing system architecture 100D includes component calculation module 135. The output of components calculation module 131 may be used as the input of component calculation module 135, and the output of calculation modules 131 and 135 may be used as the input of variance calculation module 191, and variance calculation module 191 may generate variance 192. The output of calculation modules 131 and 135 may be used as the input of standard deviation calculation module 193, and standard deviation calculation module 193 may generate standard deviation 194. Component calculation module 135 generally includes w=p−v component calculation modules for indirectly incrementally calculating w components. For example, component calculation module 135 includes calculation module 163 for indirectly incrementally calculating component Ci₁ and calculation module 164 for indirectly incrementally calculating component Ci_(w), and there are w−2 component calculation modules between them. Indirectly incrementally calculating w components includes indirectly incrementally calculating each of the w components one by one. Indirectly incrementally calculating a component includes accessing and using one or more components other than the component itself. The one or more components may have been initialized, directly incrementally calculated or indirectly incrementally calculated.

Referring to computing system architecture 100D, once all p(p=v+w) components have been calculated, variance calculation module 191 may be used for calculating a variance 192 as needed and standard deviation calculation module 193 may be used for calculating a standard deviation 194 as needed based on one or more incrementally calculated components. Variance 192 may be either a sample variance or a population variance. Standard deviation 194 may be either a sample standard deviation or a population standard deviation.

FIG. 2 illustrates a flow chart of an example method 200 for incrementally calculating a variance and/or a standard deviation for Big Data or streamed data. Method 200 will be described with respect to the components and data of computing system architectures 100A, 100B, 100C and 100D.

Method 200 includes initializing computation subset size counter 118 and v(1≤v≤p, p≥1) components of a variance and/or a standard deviation (201). For example, initialization module 132 may initialize component Cd₁ 141 with contribution 151, and initialization module 138 may initialize component Cd_(v) 145 with contribution 181. Contribution 151 and contribution 181 may be calculated from the data elements in the computation subset or from other variance and/or standard deviation calculation results such as iterative or decremental variance and/or standard deviation calculation passed in by reset module 129.

Method 200 includes accessing or receiving a data element to be added to the computation subset (202). For example, data element 101 may be accessed or received. Method 200 includes storing the accessed or received data element into an optional data buffer as needed (203). Incremental variance and/or standard deviation calculation algorithms don't need to store the accessed or received data element; however when incremental variance and/or standard deviation calculation combined with iterative variance and/or standard deviation calculation, the accessed or received data element need to be stored, so the accessed or received data element may be stored into a data buffer as needed. For example, data element 101 may be stored into an optional data buffer 125 as needed. Method 200 includes modifying the computation subset by adding the accessed or received data element to the computation subset and modifying the computation subset size counter by adding its value by 1 (204). For example, data element 101 may be added to the computation subset and computation subset size counter 118 may be modified by increasing its current value by 1 upon accessing or receiving the data element 101.

Method 200 includes directly incrementally calculating v(1≤v≤p) components of a variance and/or a standard deviation for the modified computation subset based on based on v components for the previous computation subset (205). For example, incremental algorithm 133 may be used for directly incrementally calculating component Cd₁ 143 based on component Cd₁ 141 and data element 101, and incremental algorithm 139 may be used for directly incrementally calculating component Cd_(v) 147 based on component Cd_(v) 145 and data element 101.

Directly incrementally calculating v components of a variance and/or a standard deviation for the modified computation subset includes accessing the v components of the variance and/or the standard deviation in the previous computation subset (207). For example, incremental algorithm 133 may access component Cd₁ 141. Similarly, incremental algorithm 139 may access component Cd_(v) 145.

Directly incrementally calculating v components of a variance and/or a standard deviation for the modified computation subset includes adding a contribution of the added data element to the v components mathematically (208). For example, directly incrementally calculating component Cd₁ 143 may include contribution addition module 133A adding contribution 152 to component 141 mathematically, and directly incrementally calculating component Cd_(v) 147 may include contribution addition module 139A adding contribution 182 to component Cd_(v) 145 mathematically. Contribution 152 and 182 are contributions of data element 101.

As depicted in FIG. 1A, component Cd₁ 143 includes contribution 151 (a contribution from initialization) and contributions 152 (contributions from data elements 101). Similarly, component Cd_(v) 147 includes contribution 181 (a contribution from initialization) and contribution 182 (a contribution from data element 101).

Besides directly incrementally calculated components, some components may be indirectly incrementally calculated. When not all components are directly incrementally calculated (v<p), Method 200 includes indirectly incrementally calculating w=p−v components one by one based on one or more components other than the component itself (209). Since the w indirectly incrementally calculated components are calculated using other components, they might not need to be initialized. In addition, the w indirectly incrementally calculated components might not need to be calculated in every iteration (i.e., when a data element is added to the computation subset). They only need to be calculated when a variance and/or a standard deviation are accessed to save computation time. Referring to FIG. 1B, for example, in computing system architecture 100B, component Ci₁ may be calculated based on one or more components other than Ci₁, and the one or more components may be initialized, directly incrementally calculated or indirectly incrementally calculated. Similarly, component Ci_(w) may be calculated based on one or more components other than Ci_(w), and the one or more components may have been initialized, directly incrementally calculated or indirectly incrementally calculated. Components ranging from Ci₁ to Ci_(w) only need to be calculated when a variance and/or a standard deviation are accessed. For a given component, it is possible that it is directly incrementally calculated in one algorithm but indirectly incrementally calculated in another algorithm. For example, referring to FIG. 4C, in incremental algorithm 1, component vs_(k+1) for the k+1^(th) iteration is directly incrementally calculated based on component vs_(k) for the k^(th) iteration, other components and the data element added to the computation subset. However, referring to FIG. 4E, in incremental algorithm 3, component vs_(k+1) for the k+1^(th) iteration is indirectly incrementally calculated based on components other than vs. For a given algorithm, directly incrementally calculated components must be calculated whenever there is a change in the computation subset (e.g., a data element is added to the computation subset), no matter if a variance and/or a standard deviation is accessed or not; however, an indirectly incrementally component only needs to be calculated as needed, i.e., when a variance and/or a standard deviation is accessed. For example, referring to FIG. 4E, in incremental algorithm 3, component vs_(k+1) only needs be calculated when sample variance vs_(k+1) is accessed or sample standard deviation s_(k+1) is accessed.

Method 200 includes calculating a variance as needed based on one or more incrementally calculated components of the variance (210).

Method 200 includes calculating a standard deviation as needed based on one or more incrementally calculated components of the standard deviation (211).

202-208 may be repeated as additional data elements are accessed or received. 210 and 211 may be repeated as needed. For example, subsequent to calculating components ranging from Cd₁ 143 to Cd_(v) 147, data element 102 may be accessed or received.

Incremental algorithm 133 may be used for directly incrementally calculating component Cd₁ 144 based on component Cd₁ 143. Incremental algorithm 133 may access component Cd₁ 143. Directly incrementally calculating component Cd₁ 144 may include contribution addition module 133A adding contribution 153 to component Cd₁ 143 mathematically. Contribution 153 is a contribution from data element 102.

Similarly, incremental algorithm 139 may be used for directly incrementally calculating component Cd_(v) 148 based on component Cd_(v) 147. Incremental algorithm 139 may access component Cd_(v) 147. Directly incrementally calculating component Cd_(v) 148 may include contribution addition module 139A adding contribution 183 to component Cd_(v) 147 mathematically. Contribution 183 is a contribution from data element 102.

As depicted, component Cd₁ 144 includes contribution 151 (a contribution from initialization), contribution 152 (a contribution from data element 101), and contribution 153 (a contribution from data element 102), and component Cd_(v) 148 includes contribution 181 (a contribution from initialization), contribution 182 (a contribution from data element 101), and contribution 183 (a contribution from data element 102).

When a next data element is accessed or received, component Cd₁ 144 may be used for directly incrementally calculating a component Cd₁ for the modified computation subset and component Cd_(v) 148 may be used for directly incrementally calculating a component Cd_(v) for the modified computation subset.

As depicted, reset 212 may be used for resetting incremental variance and standard deviation calculation. When reset 212 is invoked after 205, 210 or 211, the computation subset size counter and v(1≤v≤p) components of a variance and/or a standard deviation will be initialized. For example, component Cd₁ 141 may be initialized as zero when the computation subset size counter is reset to zero or a specific value if the value has already been calculated when the computation subset size counter is non-zero. The latter case may happen when combining incremental variance and standard deviation calculation with iterative/decremental variance and standard deviation calculation or decremental variance and standard deviation calculation. Component Cd_(v) 145 may be initialized in the same way.

FIG. 3A illustrates data that is accessed from a computation window 300A for incrementally calculating a variance and/or a standard deviation on streamed data. The difference between a computation window and a computation subset is that the data elements in a computation window are ordered (e.g., a data element is always added to either the right end or the left end of a computation window). For example, referring to FIG. 3A, a newly accessed or received data element is always added to the right end of computation window 300A. Computation window 300A may be either empty from very beginning or non-empty where v(1≤v≤p) components have already been calculated. As time progresses, data elements, for example, data element x₁, then x₂, then x₃, . . . , then x_(n), x_(n+1), . . . is added to the computation window and accessed respectively for incrementally calculating v components of a variance and/or a standard deviation for the modified computation subset, indirectly incrementally calculating w=p−v components, and calculating the variance and/or the standard deviation using one or more incrementally calculated components. The v components may be directly incrementally calculated based on the added data element and the v components for the previous computation window. For a given incremental algorithm, is a constant, so the number of operations for directly incrementally calculating v components is a constant, and the number of operations for indirectly incrementally calculating w=p−v components is also a constant. Thus, the computation workload for calculating p components for a given computation window with size n is reduced and a variance and/or a standard deviation may be calculated using one or more incrementally calculated components. As n increases the reduction in computation workload becomes more substantial.

FIG. 3B illustrates data that is accessed from a computation subset 300B for incrementally calculating a variance and/or a standard deviation on Big Data. The difference between a computation subset and a computation window is that the data elements in a computation subset are not ordered (e.g., a data element may be added to any position of a computation subset). For example, referring to FIG. 3B, a newly accessed or received data element may be added to any position of computation subset 300B. Computation subset 300B may be either empty from very beginning or non-empty where v(1≤v≤p) components have already been calculated. As time progresses, a data element x_(a) is added to the computation subset at the position indicated by a shaded area and accessed respectively to incrementally calculate v components of a variance and/or a standard deviation for the modified computation subset, indirectly incrementally calculating w=p−v components, and calculating the variance and/or the standard deviation based on one or more incrementally calculated components. The v components may be directly incrementally calculated based on the added data element and the v components for the previous computation subset. For a given incremental algorithm, v is a constant, so the number of operations for directly incrementally calculating v components is a constant, and the number of operations for indirectly incrementally calculating w=p−v components is also a constant. Thus, the computation workload for calculating p components for a given computation subset with size n is reduced and a variance and/or a standard deviation may be calculated using one or more incrementally calculated components. As n increases the reduction in computation workload becomes more substantial.

Embodiments of the present invention work on either streamed data or Big Data without the need to change an incremental variance and/or standard deviation algorithm. For example, three example incremental variance and/or standard deviation algorithms presented in this patent disclosure may work on either streamed data or Big Data.

FIG. 4A illustrates the definitions of a variance and a standard deviation. Suppose a computation subset X={x_(i)|i=1, . . . , n} is a subset of a Big Data set or streamed data which contains the data elements to be involved in variance and/or standard deviation calculation. Equation 401 is a traditional equation for calculating a sum S_(k) of all the data elements in computation subset X. Equation 402 is a traditional equation for calculating a mean x _(k) of all the data elements in computation subset X. Equation 403 is a traditional equation for calculating a sample variance vs_(k) for computation subset X. Equation 404 is a traditional equation for calculating a population variance vp_(k) for computation subset X. Equation 405 is a traditional equation for calculating a sample standard deviation s_(k) for computation subset X. Equation 406 is a traditional equation for calculating a population standard deviation σ_(k) for computation subset X. Suppose X has changed after some time period, say a data element x_(a) is added to computation subset X. Whenever a data element is added, the computation subset is considered as a modified computation subset. A new iteration of calculation is started each time any component of a variance and/or a standard deviation is recalculated due to a data change in the computation subset. Now, suppose components vs_(k), vp_(k), s_(k) or σ_(k) have already been calculated and the variance and/or standard deviation of the computation subset X needs to be calculated again after a data element x_(a) has been added to X. Define the modified computation subset as X′. The modified computation subset size becomes n+1. Equation 407 is a traditional equation for calculating a sum S_(k+1) of all the data elements in X′. Equation 408 is a traditional equation for calculating a mean x _(k+1) of all the data elements in X′. Equation 409 is a traditional equation for calculating a sample variance vs_(k+1) for computation subset X′. Equation 410 is a traditional equation for calculating a population variance vp_(k+1) for computation subset X′. Equation 411 is a traditional equation for calculating a sample standard deviation s_(k+1) for computation subset X′. Equation 412 is a traditional equation for calculating a population standard deviation σ_(k+1) for computation subset X′.

FIG. 4B illustrates some example components of a variance and a standard deviation and basic incremental component calculation equations used by different incremental algorithms. For example,

$\frac{1}{n - 1}{\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k}} \right)^{2}}$ may be considered as a component for a sample variance and a sample standard deviation

$\frac{1}{n}{\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k}} \right)^{2}}$ may be considered as a component for a population variance and a population standard deviation, and

${\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k}} \right)^{2}},{\frac{1}{n}{\sum\limits_{1}^{n}x_{i}}}$ and

$\sum\limits_{1}^{n}x_{i}^{2}$ may be considered as components for a sample variance, a sample standard deviation, a population variance and a population standard deviation. A sum

$S_{k} = {\sum\limits_{1}^{n}x_{i}}$ or a mean

${\overset{\_}{x}}_{k} = {\frac{1}{n}{\sum\limits_{1}^{n}x_{i}}}$ is a component to be used in several incremental algorithms, so two basic incremental equations for incrementally calculating a sum and a mean are illustrated in FIG. 4B. Equation 413 may be used for incrementally calculating a sum S_(k+1) of all the data elements in the modified computation subset X′. Equation 414 may be used for incrementally calculating a mean x _(k+1) of all the data elements in the modified computation subset X′. Either a sum S_(k+1) or a mean x _(k+1) will be used in all three incremental variance and/or standard deviation calculation algorithms described later.

FIG. 4C illustrates the first example incremental variance and/or standard deviation calculation algorithm (incremental algorithm 1) for incrementally calculating a sample variance and a sample standard deviation based on incrementally calculated sample variance and incrementally calculating a population variance and a population standard deviation based on incrementally calculated population variance. Equation 413 may be used for directly incrementally calculate a sum S_(k+1) of the data elements in the modified computation subset X′ and equation 414 may be used for directly incrementally calculate a mean x _(k+1) of the data elements in the modified computation subset X′. Equation 415 may be used for directly incrementally calculating a sample variance vs_(k+1) for the modified computation subset X′ using S_(k) or x _(k), S_(k+1) or x _(k+1) and vs_(k). Equation 416 may be used for indirectly incrementally calculating a sample standard deviation s_(k+1) for the modified computation subset X′ once vs_(k+1) is calculated. Equation 417 may be used for directly incrementally calculating a population variance vp_(k+1) for the modified computation subset X′ using S_(k) or x _(k), S_(k+1) or x _(k+1) and vp_(k). Equation 418 may be used for indirectly incrementally calculate a population standard deviation σ_(k+1) for the modified computation subset X′ once vp_(k+1) is calculated.

FIG. 4D illustrates the second example incremental variance and/or standard deviation calculation algorithm (incremental algorithm 2) for incrementally calculating a sample variance, a population variance, a sample standard deviation and a population standard deviation based on incrementally calculated components S_(k) or x _(k), S_(k+1) or x _(k+1) and SSD_(k+1). Equation 413 may be used for directly incrementally calculate a sum S_(k+1) of the data elements in the modified computation subset X′ and equation 414 may be used for directly incrementally calculate a mean x _(k+1) of the data elements in the modified computation subset X′. Equation 419 is a traditional equation for calculating component SSD_(k) for the computation subset X. Equation 420 is a traditional equation for calculating component SSD_(k+1) for the modified computation subset X′. Equation 421 may be used for directly incrementally calculating component SSD_(k+1) once components S_(k) or x _(k), S_(k+1) or x _(k+1) and SSD_(k) are calculated. Equation 422 may be then used for indirectly incrementally calculating a sample variance vs_(k+1) for the modified computation subset X′ using component SSD_(k+1). Equation 423 may be used for indirectly incrementally calculating a population variance vp_(k+1) for the modified computation subset X′ using component SSD_(k+1). Equation 424 may be used for indirectly incrementally calculating a sample standard deviation s_(k+1) for the modified computation subset X′ using component SSD_(k+1). Equation 425 may be used for indirectly incrementally calculating a population standard deviation σ_(k+1) for the modified computation subset X′ using component SSD_(k+1).

FIG. 4E illustrates the third example incremental variance and/or standard deviation calculation algorithm (incremental algorithm 3) for incrementally calculating a sample variance, a population variance, a sample standard deviation and a population standard deviation based on incrementally calculated components S_(k+1) or x ₊₁ and SS_(k+1). Equation 413 may be used for directly incrementally calculate a sum S_(k+1) of the data elements in the modified computation subset X′ and equation 414 may be used for directly incrementally calculate a mean x _(k+1) of the data elements in the modified computation subset X′. Equation 426 is a traditional equation for calculating component SS_(k) for the computation subset X. Equation 427 is a traditional equation for calculating component SS_(k+1) for the modified computation subset X′. Equation 428 may be used for directly incrementally calculating component SS_(k+1) for the modified computation subset X′ once component SS_(k) for the computation subset X is calculated. Equation 429 may be used for indirectly incrementally calculating a sample variance vs_(k+1) for the modified computation subset X′ using components S_(k+1) and/or x _(k+1) and SS_(k+1). Equation 430 may be used for indirectly incrementally calculating a population variance vp_(k+1) for the modified computation subset X′ using components S_(k+1) and/or x ₊₁ and SS_(k+1). Equation 431 may be used for indirectly incrementally calculating a sample standard deviation s_(k+1) for the modified computation subset X′ using components S_(k+1) and/or x ₊₁ and SS_(k+1) or vs_(k+1). Equation 432 may be used for indirectly incrementally calculating a population standard deviation σ_(k+1) for the modified computation subset X′ using components S_(k+1) and/or x ₊₁ and SS_(k+1) or vp_(k+1).

To demonstrate incremental variance and/or standard deviation calculation algorithms and their comparison against traditional algorithms, three examples are given below. Three computation subsets of data elements are used. For traditional algorithms, the calculations for all three computation subsets are exactly the same. For incremental algorithms, initialization of one or more components is performed for the first computation subset, and incremental calculations are performed for the second and third computation subsets.

FIG. 5A illustrates an example of calculating a sample variance and a sample standard deviation for Big Data Set 501 using traditional algorithms. Computation subset size 503 (n) is 8. Computation subset 502 includes 8 data elements in Big Data Set 501. For each computation subset, equation 402 is used for calculating a mean, then equation 403 is used for calculating a sample variance, and finally equation 405 is used for calculating a sample standard deviation. For example, for computation subset 502 the mean is calculated to be 4.75. Calculating the mean includes 1 division operation and 7 addition operations. Using the mean, the sample variance is calculated to be 7.9285714285714285714286. Calculating the sample variance includes 1 division, 8 multiplications, 7 additions and 9 subtractions. Thus, the total number of operations includes 2 divisions, 8 multiplications, 14 additions, and 9 subtractions for calculating the sample variance. Calculating the sample standard deviation includes an additional square root operation.

The same equations may be used for calculating the sample variance and the sample standard deviation for computation subset 504 as shown in FIG. 5A Cont'd 1. Computation subset 504 contains one more data element x_(a) from Big Data set 501 than computation subset 502. Computation subset size 505 becomes 9. Thus, there are more operations involved in the sample variance and sample standard deviation calculation. The calculation includes 2 divisions, 9 multiplications, 16 additions, and 10 subtractions for calculating the sample variance. Calculating the sample standard deviation includes an additional square root operation.

The same equations may be used for calculating the sample variance and the sample standard deviation for computation subset 506 as shown in FIG. 5A Cont'd 2. Computation subset 506 contains one more data element x_(a) from Big Data set 501 than computation subset 504. Computation subset size 507 becomes 10. Thus, there are more operations involved in the sample variance and sample standard deviation calculation. The calculation includes 2 divisions, 10 multiplications, 18 additions, and 11 subtractions for calculating the sample variance. Calculating the sample standard deviation includes an additional square root operation. For a computation subset with size n, traditional algorithms typically take 2 divisions, n multiplications, 2(n−1) additions and n+1 subtractions for calculating a sample variance and with an additional square root for calculating a sample standard deviation without any optimization.

FIG. 5B illustrates an example of calculating a sample variance and a sample standard deviation using the first example incremental algorithm. A mean instead of a sum is used in this example. The calculations for computation subset 502 are essentially the same as shown in FIG. 5A, which includes calculating the initial value of components x ₁ and vs₁. In practice, such calculation will not happen, because incremental variance and/or standard deviation calculation would either start from computation subset is empty or where those components have already been calculated when the computation subset is not empty. Using traditional algorithms to calculate those components on this non-empty computation subset here is only for the purpose of illustrating the incremental variance and/or standard deviation calculation algorithm. There are 2 divisions, 8 multiplications, 14 additions, and 9 subtractions for calculating the sample variance. Calculating the sample standard deviation includes an additional square root operation.

However, starting from computation subset 504, the sample variance and/or sample standard deviation may be incrementally calculated. Equations 414 may be used for incrementally calculating component x ₂ for the computation subset 504 based on component x ₁ for the computation subset 502. Equation 415 may be used for incrementally calculating the sample variance vs₂ using components vs₁, x ₁ and x ₂. Equation 416 may be used for incrementally calculating the sample standard deviation s₂ for computation subset 504 using vs₂. Equation 414 includes 1 division, 1 multiplication and 2 additions. Equation 415 includes 1 division, 2 multiplications, 1 addition, and 3 subtractions. Thus, the total operations include 2 divisions, 3 multiplications, 3 additions, and 3 subtractions for calculating the sample variance. Calculating the sample standard deviation using equation 416 includes an additional square root operation.

Equations 414, 415 and 416 may also be used for incrementally calculating the sample variance and the sample standard deviation for computation subset 506 from the components for computation subset 504. Though the computation subset size is increased, the computation workload remains constant. These calculations also include 2 divisions, 3 multiplications, 3 additions, and 3 subtractions for calculating the sample variance and with 1 additional square root for calculating the sample standard deviation. As such, the number of operations used when incrementally calculating the variance and the standard deviation is (potentially substantially) less than when using traditional equations.

FIG. 5C illustrates an example of calculating a sample variance and a sample standard deviation using the second example incremental algorithm. A mean instead of a sum is used in this example. The calculations for computation subset 502 use traditional equations to calculate the initial values of components x ₁ and SSD₁. In practice, such calculation will not happen, because incremental variance and/or standard deviation calculation would either start from computation subset is empty or where those components have already been calculated when the computation subset is not empty. Using traditional algorithms to calculate those components on this non-empty computation subset here is only for the purpose of illustrating the incremental variance and/or standard deviation calculation algorithm. Equation 402 may be used for calculating component x ₁. Equation 419 may be used for calculating component SSD₁. There are 2 divisions, 8 multiplications, 14 additions, and 9 subtractions for calculating the sample variance. Calculating the sample standard deviation includes an additional square root operation.

However, starting from computation subset 504, the sample variance and/or sample standard deviation may be incrementally calculated. Equations 414 may be used for incrementally calculating component x ₂ for the computation subset 504 based on component x ₁ for the computation subset 502. Equation 421 may be used for incrementally calculating component SSD₂ using components SSD₁, x ₁ and x ₂. Equation 422 may be used for incrementally calculating the sample variance vs₂ using component SSD₂. Equation 424 may be used for incrementally calculating sample standard deviation s₂ for computation subset 504 using component vs₂. Equation 414 includes 1 division, 1 multiplication and 2 additions. Equation 421 includes 1 multiplication, 1 addition, and 2 subtractions. Equation 422 includes 1 division. Thus, the total operations include 2 divisions, 2 multiplications, 3 additions, and 2 subtractions for calculating the sample variance vs₂. Calculating the sample standard deviation s₂ using equation 424 includes an additional square root operation.

Equations 414, 421, 422 and 424 may also be used for incrementally calculating the sample variance and the sample standard deviation for computation subset 506 from the components for computation subset 504. Though the computation subset size is increased, the computation workload remains constant. These calculations also include 2 divisions, 2 multiplications, 3 additions, and 2 subtractions for calculating the sample variance and with 1 additional square root for calculating the sample standard deviation. As such, the number of operations used when incrementally calculating the variance and the standard deviation is (potentially substantially) less than when using traditional equations.

FIG. 5D illustrates an example of calculating a sample variance and a sample standard deviation using the third example incremental algorithm. A mean instead of a sum is used in this example. The calculations for computation subset 502 use traditional equations to calculate the initial values of components x ₁ and SS₁. In practice, such calculation will not happen, because incremental variance and/or standard deviation calculation would either start from computation subset is empty or where those components have already been calculated when the computation subset is not empty. Using traditional algorithms to calculate those components on this non-empty computation subset here is only for the purpose of illustrating the incremental variance and/or standard deviation calculation algorithm. Equation 402 may be used for calculating component x ₁, and equation 426 may be used for calculating component SS₁. Equation 429 may be used for calculating the sample variance vs₁. Equation 431 may be used for calculating the sample standard deviation s₁. There are 2 divisions, 10 multiplications, 14 additions, and 2 subtractions for calculating the sample variance. Calculating the sample standard deviation includes an additional square root operation.

However, starting from computation subset 504, the sample variance and/or sample standard deviation may be incrementally calculated. Equations 414 may be used for incrementally calculating component x ₂ for the computation subset 504 based on component x ₁ for the computation subset 502. Equation 428 may be used for incrementally calculating component SS₂ for the computation subset 504 using component SS₁ for the computation subset 502. Equation 429 may be used for incrementally calculating the sample variance vs₂ using components SS₂ and Equation 431 may be used for incrementally calculating sample standard deviation s₂ for computation subset 504 using component vs₂. Equation 414 includes 1 division, 1 multiplication and 2 additions. Equation 428 includes 1 multiplication and 1 addition. Equation 429 includes 1 division, 2 multiplications, 1 addition and 1 subtraction. Thus, the total operations include 2 divisions, 4 multiplications, 4 additions, and 1 subtraction for calculating the sample variance vs₂. Calculating the sample standard deviation s₂ using equation 431 includes an additional square root operation.

Equations 414, 428, 429 and 431 may also be used for incrementally calculating the sample variance and the sample standard deviation for computation subset 506 from the components for computation subset 504. Though the computation subset size is increased, the computation workload remains constant. These calculations also include 2 divisions, 4 multiplications, 4 additions, and 1 subtraction for calculating the sample variance and with one additional square root for calculating the sample standard deviation. As such, the number of operations used when incrementally calculating the variance and the standard deviation is (potentially substantially) less than when using traditional equations.

FIGS. 5A-5B demonstrates incremental sample variance and incremental standard deviation calculations. Similar calculations for population variance and population standard deviation may also be demonstrated.

FIG. 6 illustrates computational loads for traditional sample variance algorithms and incremental sample variance algorithms for n=10. As depicted, there are fewer multiplication operations, fewer addition operations, and fewer subtraction operations using any one of the incremental algorithms.

FIG. 7 illustrates computational loads for traditional sample variance algorithms and incremental sample variance algorithms for n=1,000,000. As depicted, there are substantially fewer multiplication operations, fewer addition operations, and fewer subtraction operations using any one of the incremental algorithms.

FIG. 8 illustrates computational loads for traditional sample standard deviation algorithms and incremental sample standard deviation algorithm for n=10. As depicted, there are fewer multiplication operations, fewer addition operations, and fewer subtraction operations using any one of the incremental algorithms.

FIG. 9 illustrates computational loads for traditional sample standard deviation algorithms and incremental sample standard deviation algorithm for n=1,000,000. As depicted, there are substantially fewer multiplication operations, fewer addition operations, and fewer subtraction operations using any one of the incremental algorithms.

Similar differences in computational loads may be demonstrated for population variance and population standard deviation as well.

The present invention may be embodied in other specific forms without departing from its spirit or essential characteristics. The described embodiments are to be considered in all respects only as illustrative and not restrictive. The scope of the invention is, therefore, indicated by the appended claims rather than by the foregoing description. All changes which come within the meaning and range of equivalency of the claims are to be embraced within their scope. 

What is claimed:
 1. A computing-system-implemented method for generating a variance or a standard deviation or both for data elements in a modified computation subset, the method comprising: initializing, by a computing-device-based computing system, a sum or a mean or both for a pre-modified computation subset of a data set on at least one of one or more storage media of the computing-device-based computing system or a data stream accessible by the computing-device-based computing system, one or more other components of a variance or a standard deviation or both for the pre-modified computation subset, and a computation subset size counter n(n≥10) which specifies a number of data elements in the pre-modified computation subset; accessing or receiving, by the computing-device-based computing system, a data element to be added to the pre-modified computation subset; generating a variance or a standard deviation or both based on one or more of the initialized components; modifying, by the computing-device-based computing system, the pre-modified computation subset by: adding the accessed or received data element to the pre-modified computation subset; and modifying the computation subset size counter by increasing its value by 1; incrementally deriving, by the computing-device-based computing system, a sum or a mean or both for the modified computation subset; directly incrementally deriving, by the computing-device-based computing system and based at least in part on the one or more components of the variance or the standard deviation or both other than a sum and a mean initialized or derived for the pre-modified computation subset, one or more components of a variance or a standard deviation or both other than a sum and a mean for the modified computation subset, wherein the incrementally deriving includes: accessing the one or more components of the variance or the standard deviation or both other than a sum and a mean for the pre-modified computation subset without accessing all data elements in the modified computation subset to reduce data accessing latency thereby saving computing resources and reducing the computing-device-based computing system's power consumption; and adding any contribution of the added data element to each of the accessed components mathematically without using all data elements in the modified computation subset to reduce the number of operations performed by the computing-device-based computing system thereby increasing calculation efficiency; and generating, by the computing-device-based computing system, a variance or a standard deviation or both for the modified computation subset based on one or more of the incrementally derived components.
 2. The computing-system-implemented method of claim 1, wherein the generating a variance or a standard deviation or both further comprises indirectly incrementally deriving, by the computing-device-based computing system, one or more components of the variance or the standard deviation or both for the modified computation subset, wherein the indirectly incrementally deriving the one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 3. The computing-system-implemented method of claim 1, wherein generating a variance or a standard deviation or both comprises generating a sample variance or a sample standard deviation or both.
 4. The computing-system-implemented method of claim 1, wherein generating a variance or a standard deviation or both comprises generating a population variance or a population standard deviation or both.
 5. The computing-system-implemented method of claim 1, wherein accessing or receiving a data element to be added to the pre-modified computation subset includes accessing or receiving a plurality of z(z>1) data elements to be added to the pre-modified computation subset, and wherein the method further comprises performing, for each of the respective z data elements, the modifying the pre-modified computation subset, the incrementally deriving a sum or a mean or both for the pre-modified computation subset, the directly incrementally deriving the one or more components of a variance or a standard deviation or both for the modified computation subset, and the generating a variance or a standard deviation or both for the modified computation subset.
 6. The computing-system-implemented method of claim 1, wherein accessing or receiving a data element to be added to the pre-modified computation subset includes accessing or receiving a plurality of z(z>1) data elements to be added to the pre-modified computation subset, and wherein the method further comprises performing, for each of the respective z data elements, the modifying the pre-modified computation subset, the incrementally deriving a sum or a mean or both for the pre-modified computation subset, and the directly incrementally deriving the one or more components of a variance or a standard deviation or both for the modified computation subset.
 7. A computing system, the computing system comprising: one or more computing devices; each computing device comprising one or more processors; one or more storage media; and one or more calculation modules that, when executed by at least one of the one or more computing devices, determine a variance or a standard deviation or both for a modified computation subset, the one or more calculation modules configured to: a. initialize a computation subset size counter n(n≥10), a sum or a mean or both for a pre-modified computation subset of a data set on at least one of the one or more storage media or a data stream accessible by the computing system, and one or more other components of a variance or a standard deviation or both for the pre-modified computation subset, wherein the pre-modified computation subset contains n data elements from the data set or the data stream; b. access or receive a data element to be added to the pre-modified computation subset; c. generate a variance or a standard deviation or both based on one or more of the initialized components; d. modify the pre-modified computation subset by adding the accessed or received data element to the pre-modified computation subset and modifying the computation subset size counter; e. incrementally calculate a sum, a mean or both for the modified computation subset; f. directly incrementally calculate one or more components of a variance or a standard deviation or both other than a sum and a mean for the modified computation subset based at least in part on the one or more components of the variance or the standard deviation or both other than a sum and a mean for the pre-modified computation subset, wherein incremental calculation of the one or more components includes to add any contribution of the added data element to each of the one or more components mathematically without accessing and using all data elements in the modified computation subset to reduce data access latency and the number of operations performed by the computing system, thereby increasing calculation efficiency, saving computing resources and reducing the computing system's power consumption; and g. generate a variance or a standard deviation or both for the modified computation subset based on one or more of the incrementally calculated components.
 8. The computing system of claim 7, wherein the generating a variance or a standard deviation or both for the modified computation subset further comprises indirectly incrementally calculate one or more components of the variance or the standard deviation or both for the modified computation subset, wherein indirectly incrementally calculating one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 9. The computing system of claim 7, wherein the one or more calculation modules that, when executed by at least one of the one or more computing devices, performing b, c, d, e, f, and g multiple times.
 10. The computing system of claim 7, wherein the one or more calculation modules that, when executed by at least one of the one or more computing devices, performing b, c, d, e, and f multiple times.
 11. A computing system program product for use at a configured computing system that comprises one or more computing devices and one or more storage media, each computing device comprising one or more processors, the computing system program product comprising one or more non-transitory computing-device-readable storage media having stored thereon computing-device-executable instructions that, when executed by at least one of the one or more computing devices in the configured computing system, cause the configured computing system to perform a method for generating a variance or a standard deviation or both for data elements in a modified computation subset, the method including steps to: initialize, by a configured computing system, a computation subset size counter n(n≥10), a sum or a mean or both for a pre-modified computation subset, of size n, of a data set on at least one of one or more storage media on the configured computing system or a data stream accessible by the configured computing system, and one or more other components of a variance or a standard deviation or both for the pre-modified computation subset; access or receive, by the configured computing system, a data element to be added to the pre-modified computation subset; generate, by the configured computing system, a variance or a standard deviation or both based on one or more of the initialized components; modify, by the configured computing system, the pre-modified computation subset via adding the accessed or received data element to the pre-modified computation subset and modifying the computation subset size counter; incrementally calculate, by the configured computing system, a sum or a mean or both for the modified computation subset; directly incrementally calculate, by the configured computing system, one or more components of a variance or a standard deviation or both other than a sum and a mean for the modified computation subset based at least in part on the one or more components of the variance or the standard deviation or both other than a sum and a mean initialized or calculated for the pre-modified computation subset, including to: access the one or more components initialized or calculated for the pre-modified computation subset without accessing all data elements in the modified computation subset to reduce data accessing latency thereby saving computing resources and reducing the configured computing system's power consumption; and add any contribution of the data element added to the pre-modified computation subset to each of the accessed components mathematically without using all data elements in the modified computation subset in order to reduce the number of operations performed by the configured computing system thereby increasing calculation efficiency; and generate, by the configured computing system, a variance or a standard deviation or both for the modified computation subset based on one or more of the incrementally calculated components.
 12. The computing system program product of claim 11, wherein the computing-device-executable instructions that, when executed, further cause the configured computing system to access or receive a data element to be added to the pre-modified computation subset, to modify the pre-modified computation subset, to incrementally calculate a sum or a mean or both for the modified computation subset, to directly incrementally calculate the one or more components other than a sum and a mean, and to generate a variance or a standard deviation or both for the modified computation subset for each of multiple data elements to be accessed or received.
 13. The computing system program product of claim 11, wherein the computing-device-executable instructions that, when executed, further cause the configured computing system to access or receive a data element to be added to the pre-modified computation subset, to modify the pre-modified computation subset, to incrementally calculate a sum or a mean or both for the modified computation subset, and to directly incrementally calculate the one or more components other than a sum and a mean for each of multiple data elements to be accessed or received.
 14. The computing-system-implemented method of claim 1, wherein the adding any contribution of the added data element to each of the accessed components mathematically without using all data elements in the modified computation subset comprises adding any contribution of the added data element to each of the accessed components mathematically without using any data element other than the added data element in the modified computation sub set.
 15. The computing-system-implemented method of claim 5, wherein the generating a variance or a standard deviation or both for the modified computation subset comprises generating a variance or a standard deviation or both for the modified computation subset only when the variance or the standard deviation or both are accessed.
 16. The computing-system-implemented method of claim 15, wherein the generating a variance or a standard deviation or both for the modified computation subset further comprises indirectly incrementally deriving, by the computing-device-based computing system, one or more components of the variance or the standard deviation or both for the modified computation subset, wherein the indirectly incrementally deriving the one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 17. The computing system of claim 9, wherein the performing g comprises generating a variance or a standard deviation or both for the modified computation subset only when the variance or the standard deviation or both are accessed.
 18. The computing system of claim 17, wherein the generating a variance or a standard deviation or both further comprises indirectly incrementally calculating one or more components of the variance or the standard deviation or both for the modified computation subset, wherein indirectly incrementally calculating one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 19. The computing system program product of claim 12, wherein the generating a variance or a standard deviation or both for the modified computation subset comprises generating a variance or a standard deviation or both for the modified computation subset only when the variance or the standard deviation or both are accessed.
 20. The computing system program product of claim 19, wherein the generating a variance or a standard deviation or both for the modified computation subset further comprises indirectly incrementally calculating, by the configured computing system, one or more components of the variance or the standard deviation or both for the modified computation subset, wherein indirectly incrementally calculating the one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component. 